You are on the exactly right track. Use formula variables for readability.
var date CompletionDateBasis =
If(
Contains ([Task], "Ref Cases: Request quotation"), WeekdayAdd([Completion Date],14),
Contains ([Task],"Ref Cases: Issue PO Upload"), WeekdayAdd([Completion Date],12),
Contains ([Task],"Ref Cases: Manufacture begins"), WeekdayAdd([Completion Date],9)
);
var date InStoreStartDateBasis =
If(
Contains ([Task], "Ref Cases: Request quotation"), WeekdayAdd([In Store Start Date],14),
Contains ([Task],"Ref Cases: Issue PO Upload"), WeekdayAdd([In Store Start Date],12),
Contains ([Task],"Ref Cases: Manufacture begins"), WeekdayAdd([In Store Start Date],9)
);
IF([Dev Type]="New", $CompletionDateBasis, $InStoreStartDateBasis )
After defining a formula variable with a type and a name (names only may contain A-Z in either upper or lower case, but no spaces or numbers or special characters), and ending in a semi colon, you late refer to them with a $ so Quick Base knows that you are referring to the formula variable which you defined above.